Watch the video ... Routh is seen calmly walking backward with his shirt lifted over his head as sheriff's deputies holding guns shout instructions at him. The suspect barely utters a word while following directions to a T ... keeping his composure as he surrenders to the authorities, who slap a pair of handcuffs on him.

He even offers up his name without protest when asked by deputies to identify himself ... showing he was not rattled during the high-profile arrest.

We caught up with Sheriff Snyder on "TMZ Live" Monday, where the lawman shared that he and his deputies were initially surprised by Routh's suspiciously calm demeanor during the arrest ... though, it later reassured them they had gotten the right guy.

He added ... "I got there just right after the [traffic] stop. He was as casual as if he were going to a Sunday church service. He never asked why he was being stopped. He offered no explanation."

As TMZ previously reported ... a search was conducted in West Palm Beach, Florida on Sunday after an alleged would-be assassin posted up with an AK-style rifle in shrubbery hundreds of yards away from where Donald Trump played golf at his club.

A Secret Service agent clocked the culprit before a shot was fired ... and reportedly scared off the would-be shooter by shooting into the area -- prompting the suspect to drop the weapon and take off in a nearby SUV.

The suspect in question was later identified as Ryan Wesley Routh ... who was stopped and apprehended by the Martin County Sheriff's deputies on a Florida highway 50 miles north of the former prez's golf club.

Routh remains in custody and has been charged with possession of a firearm by a convicted felon and possession of a firearm with an obliterated serial number ... though, more charges are expected.